#89312

order	448
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #362

Subgroups

	subgroup	G/H
center	C₂	#6395
commutator	C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₇
Fitting	C₇×M₃₂	C₂
socle	C₂×C₇	#102
prime radical	C₂²×C₇	SD₁₆
cosocle	C₂×C₇×C₈	C₂²
hypercenter	M₃₂	D₇
hypocenter	C₇	#362
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,4,6,7,5,3)(8,10,11,23,24,35,30,36,13,20,26,19,12,21,25,39)(9,14,15,31,27,33,38,34,17,28,32,18,16,29,22,37);(2,3)(4,5)(6,7)(8,9,13,17)(10,18,20,31)(11,22,26,38)(12,27,24,16)(14,23,28,19)(15,30,32,25)(21,37,35,34)(29,36,33,39)

Polycyclic presentation:

2^6,7|a^2=f,b^2=cd,c^2=ef,e^2=f,b^a=bc,c^a=ce,e^a=ef,g^a=g^6,c^b=cf,d^b=df

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726844, #11726871, #11726881, …
maximal normal subgroup of	#11726844, #11726871, #11726881, …
maximal quotient of	#11725323, #11725335, #11725367, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 74);